Variable 1: Las consignas
VII. EFERENCIAS
The statistical analysis of experimental data is a fundamental task in any applied science. Statistics can—and must—be similarly applied to sets of archaeoastron-omy data. However, as we shall see, the cultural/sociological nature of data here means that we have to apply statistics in archaeoastronomy in a very prudent and responsible way (Ruggles1999).
Let us suppose to have a building which exhibits a certain number of alignments (e.g. different windows), or a set of buildings in which we have measured a potentially interesting feature (e.g. the entrance corridors of a group of tombs).
Afirst note of warning must be sounded immediately. Statistics is an extremely evolved science that provides a series of complex and sophisticated mathematical instruments for the purposes of data analysis. And yet, in archaeoastronomy, it is difficult to have datasets with more than a few dozen values; as a consequence, it is advisable to use rather elementary statistical approaches as the one I am about to describe here. A second note of caution. Sometimes, an effect called data selection can undermine any statistical reasoning right from the outset. To avoid data selection one must take great care in measuring and considering any possible alignment at a site (for instance, between stones in a megalithic monument), not just choosing those which are likely to have an astronomical interpretation. Similarly, one should preferably decide a priori which astronomical targets might have been of interest for the builders, and not a posteriori. It is, in fact, easy to match a certain azimuth with a star, especially if one juggles around with the allowed error, or with uncertainties in the date of construction, or with both (see also Sect.6.2). All in all, the definition of targets of potential interest must be based on reasonable cultural arguments. A (non-exhaustive) list of targets to be considered as eligible might be the following:
– the four cardinal directions – the two solstitial axes
– the arc of the Sun between the solstices
– the arc of the Sun climbing in the sky (from the azimuth of the winter solstice to due south)
– the major lunar standstills
– the rising/setting of brightest stars, at least in all cases in which a cultural interest in them is soundly documented (for instance in ancient Egypt) and a reasonable date for construction is known
– slightly less obvious targets, if there are historical/cultural reasons for them.
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Examples: the Zenith passages of the Sun for the pre-Columbian cultures of Mesoamerica, the Pleiades for the Incas; Venus for the Mesoamerican cultures; the Pleiades and also faint but culturally relevant constellations such as Delphinus for the Greeks, and so on.
Once a satisfactory definition of potentially interesting targets has been agreed on, we have still to agree on the allowed error, in other words, we have to define what matching targets means. This is quite a delicate point, since there are two completely different things—confusingly, with the same name—going on. The first is the modern error, the one which is intrinsic to our measurements. If we used a magnetic compass, it can be quite substantial, as we have seen (not less than±½°).
So if the azimuth matches the target within the modern error, it, of course, matches.
But another error, which we cannot really foresee or control, is present: the original one. As a matter of fact, ancient surveyors also used an instrument (the naked eye, backed up with natural or artificial foresights) and therefore committed an error. It is thus usually necessary to have an idea of how much the original error might have been. For instance, for observations made on theflat horizon of the jungle in the tropical area of the Mayas we would not expect the original error to have been less than±1°, while we have incontrovertible proof that Roman surveyors were able to measure the meridian within 10′, and that the ancient Egyptians oriented the pyramids of Khufu and Khafra at Giza with even greater precision (to the point that measuring the side of the Khufu pyramid with a magnetic compass would be useful only if we want to use the measure to have an estimate of the error of the compass).
Once all of this has been carefully considered can wefinally seek an answer to the question: are the astronomical correspondences that have been found merely fortuitous, or do they reflect the explicit intentions of the builders? To solve this problem we can use probability. To start with, let us consider a marksman whofires randomly n times, without looking. What are the odds of him hitting exactly k targets? If p is the probability of hitting a target with a single shot, probability P is a function of n and k given by the Binomial distribution
P¼ n!
k!ðn kÞ!pkð1 pÞnk
where the“factorial” symbol ! after an integer means that the product of all positive integers less than or equal to that number must be taken, for instance, 4! ¼ 4 3 2 1 ¼ 24.
To apply binomial distribution in archaeoastronomy, suppose that in measuring a site, out of n total alignments, we have found k pointing to astronomical targets.
What are the odds that this result has come about by chance? To calculate this we mustfirst estimate the probability p of a single alignment occurring. Then, we can use the binomial distribution formula to estimate what the probability is that the result—k out of n—is down to a chance, and express the result in the form of a fraction. For example, we mayfind that there is a 1/1000 probability of getting the result by chance, or 3/100, or 1/25, or whatever. The smaller these fractions are, the
3.4 Statistical Tools 47
higher the chances for non-coincidental orientations (see e.g. Polcaro and Polcaro 2006).
Of course, naturally, if we had found a number of alignments greater than k, we would have tried to interpret this information in the same way. This means that we are not quite approaching the issue in a correct manner, as we should be considering the probability of reaching k or more targets, not exactly that. Clearly, this cumu-lative distribution will always give greater probabilities as compared to simple distribution. By adding up the single probabilities appropriately, such a distribution can be calculated relatively easily. For the sake of simplicity, I use here only the standard binomial formula; the results actually do not differ qualitatively each time the single shot probability is significantly less than, say, ½, as usually occurs in archaeoastronomical applications.
Imagine now that a chance probability has been obtained, and that this proba-bility is “small”. We have two problems. First of all, even if we accept that the orientations are definitely not coincidental, the astronomical interpretation might not be correct. A trivial example might be that of a cluster of tombs excavated in a rock wall. The common orientation clearly is roughly orthogonal to the wall, for geo-morphological reasons, but it may, by chance, coincide with an astronomical target. A much less banal example would be that of buildings orientated towards natural features of the landscape, such as, for instance, a mountain. In the first example, our analysis if no use, as the alignment is due to geo-morphological reasons, but in the latter it may reveal that the mountain was sacred, and deliber-ately selected.
Another issue is whether we have any way to give a mathematical meaning to the concept of“small” probability. For instance, to what extent are we prepared to believe in an astronomical hypothesis if the probability is 1/1000, or 3/100, or 1/25?
The answer must come from the establishment of a common agreement on the threshold which allows us to claim that a phenomenon is not random. Since this problem is shared by all the sciences, it is certainly not the job of archaeoastronomy to establish this threshold, but, rather, we should conform to the methods commonly adopted in all the other sciences. This method is based on the obvious fact that the further a result deviates from the mean, the greater the chance it will be statistically significant. To measure this distance the concept of standard deviation can be introduced.
For the binomial distribution the mean is given by the product np between the number of cases and the single probability, and the standard deviation is
r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi npð1 pÞ p
It can be shown that 68 % of the values of a binomial distribution fall within one standard deviation from the mean, 95.5 % within two standard deviations from the mean, and that as many as 99.7 % fall within three standard deviations. Therefore, a common—and quite natural—convention is to assume that data start to be signifi-cant if they are at least“at the 2σ level”—that is, their distance from the mean is greater than 2σ—and that they are at a high level of confidence if they pass 3σ.
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The threshold at 2σ is, however, by no means universal: for instance, C14 dating used in archaeology is often given only at a worrying 1σ level of confidence, while in frontier experiments in fundamental physics, levels of confidence of 5σ are some-times required. What about archaeoastronomy? As a matter of fact, seeking a 3σ level by default (as some would advocate, see e.g. Schaefer2006) is simply a way of destroying the credibility of much of this discipline, given that the data samples are usually very limited. For example, it can easily be seen that any assertion of intentional alignment at a single site (like a row of megaliths) would thus be auto-matically rejected. Indeed, even considering only the solstitial and the cardinal directions as significant, one has eight possible azimuths which, allowing 1° of error, give a probability for a chance alignment along one of them of 4.4 % (since 16/360 = 0.044), which is just below 2σ level and very far from the 3σ. Clearly then, such a drastic approach cannot be accepted. Actually—and amazingly—what hap-pens in many cases is that only alignments which would have no need for the use of statistics to be considered as intentional by any sane-of-mind person can be claimed to be accurate at the 3σ level (we shall see two examples in this book: the claims that the Greek temples of Sicily are oriented towards the rising Sun, and that the megalithic sepulchres of Iberia are oriented within the arc of the Sun’s rising/climbing).
To conclude, the statistical approach must always be considered with reference to the special characteristics of archaeoastronomical data: a physicist who is unconvinced of his experimental results at 2σ level can repeat the experiment to improve statistical analysis, but we cannot erect new megalithic tombs to extend our samples. Of course, on the other hand, iniquitous use of statistics through data selection, target selection and the like must be strenuously avoided. Finally, as far as single sites are concerned, I personally would argue for the right for archaeoastronomy to study single, unique sites and to come up with some very reasonable conclusions, with or without anyσ level that might be applicable. We shall see examples of such one-off sites in this book, and three of them are—I repeat, at least in my view—among the most outstanding successes of archaeoas-tronomy in absolute, as well as being among the most beautiful and complex monuments ever created by the human mind: the Pyramid of Khufu at Giza, the Castillo at Chichen Itza’ and the Pantheon in Rome.
References
Aveni, A. F. and G. Romano. 2001. Temple Orientation in Magna Grecia and Sicily. Journal for the History of Astronomy 31: 52–57.
Hoskin, M. 1998 Studies in Iberian Archaeoastronomy: (5) Orientations of Megalithic Tombs of Northern and Western Iberia Journal for the History of Astronomy, Archaeoastronomy Supplement, Vol. 29, p. S39.
Polcaro A. and Polcaro V. (2006) Early Bronze Age Dolmens in Jordan and their orientations Mediterranean Archaeology and Archaeometry 6, pp. 169–174.
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Ruggles C.L.N. (1999). Astronomy In Prehistoric Britain And Ireland. New Haven & London:
Yale University Press.
Schaefer, B. (2006). Case Studies of the Three Most Famous Claimed Archaeoastronomical Alignments in North America. In T. Bostwick, & B. Bates, Viewing the Sky Through Past and Present Cultures: Selected Papers from the Oxford VII International Conference on Archaeoastronomy (pp. 27–56). Phoenix: City of Phoenix Parks and Recreation Department.
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